"360 WESTLEY RICHARDS RIMLESS NITRO EXPRESS
This appears to be a rimless version of the 400/360 WESTLEY RICHARDS NITRO EXPRESS. It was listed in the 1912 Westley Richards catalogue and designed to be used in a bolt action magazine rifle. The headstamp is right on the edge, so it seems that existing case were used where the rim was turned down and the extractor groove added. It might have been some experimental following the European tradition to have a rimmed and rimless cartridge in the same caliber."

Another reference:
ammo-one.com

"The 400/360 Nitro Express (2.75" case) was introduced in the very early 1900s. Westley Richards, Fraser, and Evans versions have .358 to .360" diameter projectiles. The Purdey version used a .364 to .367" projectile. Westley Richards also made a rimless version."